1. Field of the Invention
The present invention relates to a system and method for manufacturing a semiconductor device, and more particularly, to a system for analyzing a mask topography and a method of forming an image using the system.
2. Description of the Related Art
In photolithography, as a feature size on a mask becomes smaller than the wavelength of light used to illuminate the mask, the aspect ratio of the thickness of the mask to the feature size becomes higher. Accordingly, it is difficult to accurately simulate a topography of a pattern drawn on the mask (referred to as a “mask topography” hereinafter) using a conventional Kirchhoff method that assumes the mask to be a thin mask without topographic features.
The conventional Kirchhoff method will now be explained with reference to FIGS. 1A and 1B. Light transmitted through a space between light blocking layers 20 disposed on a substrate 10 had no phase change and had a higher transmittance or amplitude-like step function. An interval between the light blocking layers 20 on a 4× mask was 140 nm, and the wavelength of the light was 280 nm. In detail, FIG. 1A illustrates a plot of an amplitude (solid line) and a phase (dotted line) according to a horizontal position or distance in nanometers (nm). The amplitude was constantly high only in an area where the light blocking layers 20 were not formed, and there was no phase change according to the distance. Also, FIG. 1B illustrates an amplitude (arrow) and a phase according to frequency (f) on a pupil surface of a projection lens, wherein zero order light had an amplitude of a1 and ± first order light had an amplitude of a2 without any phase change.
To more accurately simulate a topography, an electromagnetic (EM) field analysis method has been proposed that calculates a transmittance and a phase of an electric field (E-field) on a mask surface considering a mask topography effect.
The conventional EM field analysis method will now be explained with reference to FIGS. 2A and 2B. Light transmitted through a space between light blocking layers 20 disposed on a substrate 10 has an amplitude or transmittance and a phase which are different from those of the conventional Kirchhoff method. Referring to an image 30 of FIG. 2A illustrating an amplitude and a phase and a graph 40 illustrating an amplitude (solid line) and a phase (dotted line) according to a horizontal position or distance, the light had an amplitude shaped like a distorted step function, and a variation in phase such that the phase below the light blocking layer 20 was changed.
Referring to FIG. 2B illustrating an amplitude (arrow) according to frequency (f) on a pupil surface of a projection lens, the amplitudes of zero order light and first order light were higher than their counterparts a1 and a2 of FIG. 1B, respectively. Also, the phase of the first order light was 10° or so. Briefly, a change in the amplitude and the phase of an E-field on a mask surface results in a change in the amplitude and the phase of an E-field of zero order light and first order light on a pupil surface of a projection lens. The change in the amplitude and the phase of the E-field on the pupil surface of the projection lens changes the contrast of an image on a wafer and affects focus tilt. Accordingly, the EM field analysis method can calculate the E-field on the mask surface and thus can accurately predict the image produced on the wafer.
When the E-field on the mask surface is calculated using the EM field analysis method and a mask pattern is a line and space (referred to as L/S) pattern, a calculation time is 1 minute or less. However, when the mask pattern is a complex two-dimensional (2D), pattern, a calculation time is increased exponentially. In particular, it is virtually impossible to calculate a complex 2D pattern in design for manufacturability (DFM) or optical proximity correction (OPC). Accordingly, attempts have been made to reduce a time required to calculate the amplitude or transmittance and the phase of an E-field on a mask surface by using an approximation model instead of the EM field analysis method.
Table 1 shows the characteristics of the conventional Kirchhoff method, the EM field analysis method, and the approximation model. Here, a domain decomposition method (DDM) and a boundary layer method were used as examples of the approximation model. The methods were compared in terms of whether they can calculate a mask topography effect, calculation time, and calculation accuracy.
TABLE 1Possibility ofcalculating maskCalculationCalculationMethodtopography effecttime (speed)accuracyKirchhoffImpossible1timePoorEM field analysisPossible10~100timeGoodDDMPossible2~5timesBetween goodand fairBoundary layerPossible1timeFair
Referring to Table 1, the EM field analysis method had the best calculation accuracy, but had the longest calculation time as, described above. The DDM method had a relatively good calculation accuracy, but had a long calculation time. In terms of the calculation time, the boundary layer method was the best for DFM and OPC. However, it was difficult to implement the boundary layer method and thus desired boundary layers could not be used for various patterns. Also, since the boundary layer method used only one predetermined boundary layer, the calculation accuracy was not high.